R05
Code No: R05210201
Set No. 2
√ 1+z z 1−2xz +z 2 d dx
(b) Prove that
−
1 z
=
∞ P
(Pn (x) + Pn+1 (x)) z n .
n=0
2 ). (xJn Jn+1 ) = x(Jn2 − Jn+1
ld .
1. (a) Prove that
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II B.Tech I Semester Examinations,MAY 2011 MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????
C
or
(c) Prove that cos x=J0 –2J2 + 2J4+............ [6+5+5] R 2 dz 2. (a) Evaluate (z −2z−2) where c is | z − i | = 1/2 using Cauchy’s integral for(z 2 +1)2 z c mula. R (b) Evaluate (z 2 + 3z + 2) dz where C is the arc of the cycloid x = a(θ + sin θ), y = a (1 − cos θ) between the points (0,0) to (πa, 2a). 3. Expand
1 z(z 2 −3z+2)
for the regions
(a) 0 < |z| < 1
uW
(b) 1 < |z| < 2
[8+8]
(c) |z| > 2.
[16]
4. Evaluate the following using β − Γ functions. (a)
R1
(x log x)3 dx
0 π/2 R
sin11 θ cos3 θdθ
nt
(b)
0
(c)
R∞
2
x6 e−3x dx.
0
[5+5+6]
Aj
5. (a) Find the analytic function f(z) = u+iv if u–v= f (π/2) = 3−i . 2
ey − cos x+sin x cosh y−cos x
given that
(b) Find the principal values of (1+i)(1−i) .
6. (a) Evaluate (b) Evaluate
R2π
0 R∞ 0
dθ (5−3 sin θ)2
x sin mx dx x4 +16
[8+8]
using residue theorem.
using residue theorem.
7. (a) Under the transformation w=1/z, find the image of the circle |z-2i|=2. 1
[8+8]
R05
Code No: R05210201
(b) Under the transformation w = w-plane.
z−i , 1−iz
Set No. 2
find the image of the circle |z|=1 in the [8+8]
8. (a) Find the poles and residues at each pole of the function cosec2 z. R zeiz dz (b) Evaluate (z 2 +9)2 where c is the circle |z | = 4 by residue theorem. C
R C
z 3 dz (z−1)2 (z−3)
where c is | z | = 2 by residue theorem.
[5+5+6]
in
(c) Evaluate
Aj
nt
uW
or
ld .
?????
2
R05
Code No: R05210201
Set No. 4
in
II B.Tech I Semester Examinations,MAY 2011 MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Under the transformation w=1/z, find the image of the circle |z-2i|=2.
(b) Evaluate
0 ∞ R 0
3. Expand
dθ (5−3 sin θ)2
x sin mx dx x4 +16
1 z(z 2 −3z+2)
find the image of the circle |z|=1 in the [8+8]
using residue theorem.
using residue theorem.
[8+8]
or
R2π
2. (a) Evaluate
z−i , 1−iz
ld .
(b) Under the transformation w = w-plane.
for the regions
(a) 0 < |z| < 1 (b) 1 < |z| < 2
uW
(c) |z| > 2.
4. (a) Prove that (b) Prove that
√ 1+z z 1−2xz +z 2 d dx
−
1 z
=
∞ P
[16]
(Pn (x) + Pn+1 (x)) z n .
n=0
2 (xJn Jn+1 ) = x(Jn2 − Jn+1 ).
(c) Prove that cos x=J0 –2J2 + 2J4+............
[6+5+5]
5. Evaluate the following using β − Γ functions. R1
(x log x)3 dx
nt
(a)
0
(b)
0 ∞ R
sin11 θ cos3 θdθ 2
x6 e−3x dx.
Aj
(c)
π/2 R
0
[5+5+6]
6. (a) Find the analytic function f(z) = u+iv if u–v= f (π/2) = 3−i . 2
ey − cos x+sin x cosh y−cos x
given that
(b) Find the principal values of (1+i)(1−i) . 7. (a) Find the poles and residues at each pole of the function cosec2 z. R zeiz dz (b) Evaluate (z 2 +9)2 where c is the circle |z | = 4 by residue theorem. C
3
[8+8]
R05
Code No: R05210201 (c) Evaluate
R C
8. (a) Evaluate
R c
z 3 dz (z−1)2 (z−3)
Set No. 4
where c is | z | = 2 by residue theorem.
(z 2 −2z−2) dz (z 2 +1)2 z
[5+5+6]
where c is | z − i | = 1/2 using Cauchy’s integral for-
mula. (b) Evaluate
R
(z 2 + 3z + 2) dz where C is the arc of the cycloid x = a(θ + sin θ),
C
Aj
nt
uW
or
ld .
?????
[8+8]
in
y = a (1 − cos θ) between the points (0,0) to (πa, 2a).
4
R05
Code No: R05210201
Set No. 1
in
II B.Tech I Semester Examinations,MAY 2011 MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ????? 1. (a) Under the transformation w=1/z, find the image of the circle |z-2i|=2. z−i , 1−iz
find the image of the circle |z|=1 in the [8+8]
2. Evaluate the following using β − Γ functions. (a)
R1
(x log x)3 dx
0 π/2 R
sin11 θ cos3 θdθ
or
(b)
0
(c)
R∞
ld .
(b) Under the transformation w = w-plane.
2
x6 e−3x dx.
∞ P
uW
0
√ 1+z z 1−2xz +z 2
3. (a) Prove that
d dx
(b) Prove that
−
1 z
=
(Pn (x) + Pn+1 (x)) z n .
n=0
2 ). (xJn Jn+1 ) = x(Jn2 − Jn+1
(c) Prove that cos x=J0 –2J2 + 2J4+............ 4. (a) Evaluate
0 R∞
dθ (5−3 sin θ)2
0
x sin mx dx x4 +16
[6+5+5]
using residue theorem.
using residue theorem.
nt
(b) Evaluate
R2π
[5+5+6]
5. (a) Find the analytic function f(z) = u+iv if u–v= f (π/2) = 3−i . 2
[8+8] ey − cos x+sin x cosh y−cos x
given that
Aj
(b) Find the principal values of (1+i)(1−i) . [8+8] R 2 dz 6. (a) Evaluate (z −2z−2) where c is | z − i | = 1/2 using Cauchy’s integral for(z 2 +1)2 z c mula. R (b) Evaluate (z 2 + 3z + 2) dz where C is the arc of the cycloid x = a(θ + sin θ), C
y = a (1 − cos θ) between the points (0,0) to (πa, 2a). 7. Expand
1 z(z 2 −3z+2)
for the regions
(a) 0 < |z| < 1 5
[8+8]
R05
Code No: R05210201
Set No. 1
(b) 1 < |z| < 2 (c) |z| > 2.
[16]
8. (a) Find the poles and residues at each pole of the function cosec2 z. R zeiz dz (b) Evaluate (z 2 +9)2 where c is the circle |z | = 4 by residue theorem. C
R C
z 3 dz (z−1)2 (z−3)
where c is | z | = 2 by residue theorem.
Aj
nt
uW
or
ld .
?????
[5+5+6]
in
(c) Evaluate
6
R05
Code No: R05210201
Set No. 3
(b) Evaluate
0 R∞ 0
2. (a) Evaluate
dθ (5−3 sin θ)2
x sin mx dx x4 +16
R c
using residue theorem.
using residue theorem.
(z 2 −2z−2) dz (z 2 +1)2 z
where c is | z − i | = 1/2 using Cauchy’s integral for-
mula. (b) Evaluate
R
[8+8]
ld .
R2π
1. (a) Evaluate
in
II B.Tech I Semester Examinations,MAY 2011 MATHEMATICS-III Common to ICE, E.COMP.E, ETM, E.CONT.E, EIE, ECE, EEE Time: 3 hours Max Marks: 80 Answer any FIVE Questions All Questions carry equal marks ?????
(z 2 + 3z + 2) dz where C is the arc of the cycloid x = a(θ + sin θ),
or
C
y = a (1 − cos θ) between the points (0,0) to (πa, 2a). 3. Expand
1 z(z 2 −3z+2)
for the regions
uW
(a) 0 < |z| < 1
[8+8]
(b) 1 < |z| < 2 (c) |z| > 2.
[16]
4. (a) Under the transformation w=1/z, find the image of the circle |z-2i|=2. (b) Under the transformation w = w-plane.
z−i , 1−iz
find the image of the circle |z|=1 in the [8+8]
nt
5. Evaluate the following using β − Γ functions. (a)
R1
(x log x)3 dx
0
0 ∞ R
sin11 θ cos3 θdθ
Aj
(b)
π/2 R
(c)
2
x6 e−3x dx.
0
6. (a) Prove that (b) Prove that
[5+5+6] √ 1+z z 1−2xz +z 2 d dx
−
1 z
=
∞ P
(Pn (x) + Pn+1 (x)) z n .
n=0
2 (xJn Jn+1 ) = x(Jn2 − Jn+1 ).
(c) Prove that cos x=J0 –2J2 + 2J4+............
7
[6+5+5]
R05
Code No: R05210201
Set No. 3
7. (a) Find the analytic function f(z) = u+iv if u–v= f (π/2) = 3−i . 2
ey − cos x+sin x cosh y−cos x
given that
(b) Find the principal values of (1+i)(1−i) .
[8+8]
C
(c) Evaluate
R C
z 3 dz (z−1)2 (z−3)
where c is | z | = 2 by residue theorem.
[5+5+6]
Aj
nt
uW
or
ld .
?????
in
8. (a) Find the poles and residues at each pole of the function cosec2 z. R zeiz dz (b) Evaluate (z 2 +9)2 where c is the circle |z | = 4 by residue theorem.
8
